[OFFTOPIC] Maps and functions (was Re: Apply a function to e

[Mikael Olofsson]

On 05-Nov-99 Charles G Waldman wrote:
 >  Hmm... what kind of "map" are you thinking of that could send 2 to
 >  "both 3 and 4"?  I think such a thing is not a mapping, at least not
 >  according to any definition of "map" that I've ever seen.  If you
 >  look, for instance, in any basic topology text, you will see that
 >  "map" and "function" mean exactly the same thing, when used as nouns.
 >  "Map" also has a sense as a transitive verb, which means "to apply a
 >  function to", which is where the Python/functional programming usage
 >  comes from.

There is nothing that says that a map cannot output a set. But then again, 
that is an object (in a set of sets), even if the set consists of many 
smaller objects. As an example, the inverse mapping of the squaring 
function (a->a^2) is a mapping from the set of reals to a set of sets of 
reals, sets containing at most two real elements (b->{+/-sqrt(b)},{0}, or 
the empty set), depending on if b is positive, zero, or negative.

You are probably right when you state that a function is a mapping is
a function, but people tend to give them different interpretations.
As an example (to make things even more confused), I have a reference 
(Davenport: Probability and Random Processes) that uses the words in the 
following way: 

  A function determines a mapping from one set to another. 

This suggests that a function is just an expression, and the mapping is 
its action on the elements of a set.

Are-we-being-too-scientific-ly y'rs

/Mikael

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Date:    05-Nov-99
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